The following items are presently staged at the 40m Bake Lab (see photo indicating current location) (noting items broght by Koji as well):
The following item is in its home in Downs 303 (Modal Lab)
The 40m Bake Lab's Dirty ABO's OMEGA PID controller was borrowed for another oven in the Bake Lab, so I have had to play with the tuning and parameters to recover a suitable bake profile. This bake is pictured below (please excuse the default excel formatting).
I have increased the ramp time, temperature offset, and thermal mass within the oven; after retuning and applying the parameters indicated, the rate of heating/cooling never exceeds .5°C/min.
The ABO is controlled by a different temperature readout from the data logger used to collect data; the ABO readout is a small bead in contact with the shelf, while the data logger is a lug sandwiched between two stainless steel masses upon the shelf. I take the data logger profile to be more physically similar to the heating experienced by an optic in a gluing fixture, so I feel happy about the results of the above bake.
I plan to add the data source file to this post at my earliest convenience.
The 40m Bake Lab's Dirty ABO's OMEGA PID controller was borrowed for another oven in the Bake Lab (sound familiar? OMC elog 377), so I have had to play with the tuning and parameters to recover. This bake seemed to inadequately match the intended temperature profile for some reason (intended profile is shown by plotting prior qualifying bake for comparison).
The parameters utilized here are exactly matching the prior qualifying bake, except that the autotuning may have settled on different parameters.
Options to proceed, as I see them, are as follows:
Follow up on OMC elog 379
I was able to obtain the following (dark blue) bake profile, which I believe is adequate for our needs.
The primary change was a remounting of the thermocouple to sandwich it between two stainless steel masses. The thermocouple bead previously was 1) in air and 2) close to the oven skin.
OMC PZT Assy Production Cure Bake (ref. OMC elog 381) for PZT Assy #9 and #10 started 27 September 2019 and completed 28 September 2019. Captured in the below figure (purple trace). Raw data has been posted as an attachment as well.
We have monitored the temperature in two ways:
1) Datalogger thermocouple data (purple trace).
2) Checking in on temperature of datalogger thermocouple (lavender circles) and drive thermocouple (lavender diamonds), only during initial ramp up.
Comments on bake:
This post captures the curing timeline followed by OMC PZT Assys #9 and #10.
Source file posted in case any updates or edits need to be made.
The following is the current status of the epoxies used in assembly of the OMC (excerpt from C1900052)
Re-purchasing efforts are underway and/or complete
[Koji, Jordan, Stephen]
The beam dumps, bonded on Fri 06 Dec 2019, were placed in the newly tuned and configured small dirty ABO at the Bake Lab on Fri 13 Dec 2019.
Images are shared and references are linked below
Bonding log entry - https://nodus.ligo.caltech.edu:8081/OMC_Lab/386
Bake ticket - https://services.ligo-wa.caltech.edu/clean_and_bake/request/992/
OMC Beam Dump - https://dcc.ligo.org/LIGO-D1201285
OMC Unit 4 Build Machined Parts are currently located in Stephen's office. See image of large blue box from office, below.
Loaned item D1100855-V1-00-OMC08Q004 to Don Griffith for work in semi-clean HDS assy.
This includes mass mounting brackets, cable brackets, balance masses, etc. For full inventory, refer to ICS load Bake-9527 (mixed polymers) and Bake-9495 (mixed metals).
Inventory includes all items except cables. Plasma sprayed components with slight chipping were deemed acceptable for Unit 4 use. Cable components (including flex circuit) are ready to advance to fabrication, with a bit more planning and ID of appropriate wiring.
More explicit insights into the inventory for the Unit 4 build. Image of inventory included below.
ref: E1900034 and other associated documents.
EDIT (ZK): All the plots here were generated using my MATLAB cavity modeling tool, ArbCav. The utility description is below. The higher-order mode resonance plots are direct outputs of the function. The overlap plots were made by modifying the function to output a list of all HOM resonant frequencies, and then plotting the closest one as a function of cavity length. This was done for various values of highest mode order to consider, as described in the original entry.
This function calculates information about an arbitrary optical cavity. It can plot the cavity geometry, calculate the transmission/reflection spectrum, and generate the higher-order mode spectrum for the carrier and up to 2 sets of sidebands.
The code accepts any number of mirrors with any radius of curvature and transmission, and includes any astigmatic effects in its output.
As opposed to the previous version, which converted a limited number of cavity shapes into linear cavities before performing the calculation, this version explicitly propagates the gouy phase of the beam around each leg of the cavity, and is therefore truly able to handle an arbitrary geometry.
I expressed concern that arbitrarily choosing some maximum HOM order above which not to consider makes us vulnerable to sitting directly on a slightly-higher-order mode. At first, I figured the best way around this is to apply an appropriate weighting function to the computed HOM frequency spacing. Since this will also have some arbitrariness to it, I have decided to do it in a more straightforward way. Namely, look at the spacing for different values of the maximum mode number, nmax, and then use this extra information to better select the length.
Below are the spacing plots for the bowtie (flat-flat-curved-curved) and non-bowtie (flat-curved-flat-curved) configurations. Points on each line should be read out as "there are are no modes of order N or lower within [Y value] linewidths of the carrier TEM00 transmission", where N is the nmax appropriate for that trace. Intuitively, as more orders are included, the maxima go down, because more orders are added to the calculation.
*All calculations are done using my cavity simulation function, ArbCav. The mode spacing is calculated for each particular geometry by explicitly propagating the gouy phase through each leg of the cavity, rather than by finding an equivalent linear cavity*
Since achievable HOM rejection is only one of the criteria that should be used to choose between the two topologies, the idea is to pick one length solution for EACH topology. Basically, one maximum should be chosen for each plot, based on how how high an order we care about.
For the bowtie, the nmax = 20 maximum at L = 1.145 m is attractive, because there are no n < 20 modes within 5 linewidths, and no n < 25 modes within ~4.5 linewidths. However, this means that there are also n < 10 modes within 5 linewidths, while they could be pushed (BLUE line) to ~8.5 linewidths at the expense of proximity to n > 15 modes.
Therefore, it's probably best to pick something between the red and green maxima: 1.145 m < L < 1.152 m.
By manually inspecting the HOM spectrum for nmax = 20, it seems that L = 1.150 m is the best choice. In the HOM zoom plot below and the one to follow, the legend is as follows
Following the same logic as above, the most obvious choice for the non-bowtie is somewhere between the red maximum at 1.241 m and the magenta maximum at 1.248 m. This still allows for reasonable suppression of the n < 10 modes without sacrificing the n < 15 mode suppression completely.
Upon inspection, I suggest L = 1.246 m
I reiterate that these calculations are taking into account modes of up to n ~ 20. If there is a reason we really only care about a lower order than this, then we can do better. Otherwise, this is a nice compromise between full low-order mode isolation and not sitting directly on slightly higher modes.
One complication that arises is that all of these are highly dependent on the actual RoC of the mirrors. Unfortunately, even the quoted tolerance of ±1% makes a difference. Below is a rendering of the RED traces (nmax = 20) in the first two plots, but for R varying by ±2% (i.e., for R = 2.45 m, 2.50 m, 2.55 m).
The case for the non-bowtie only superficially seems better; the important spacing is the large one between the three highest peaks centered around 1.24 m.
Also unfortunately, this strong dependence is also true for the lowest-order modes. Below is the same two plots, but for the BLUE (nmax = 10) lines in the first plots.
Therefore, it is prudent not to pick a specific length until the precise RoC of the mirrors is measured.
Assuming the validity of looking at modes between 10 < n < 20, and that the curved mirror RoC is the design value of 2.50 m, the recommended lengths for each case are:
HOWEVER, variation within the design tolerance of the mirror RoC will change these numbers appreciably, and so the RoC should be measured before a length is firmly chosen.
EDIT 2 (ZK): As with the previous post, all plots and calculations here are done with my MATLAB cavity modeling utility, ArbCav.
EDIT (ZK): Added input q parameters for OMMT
I found the nice result that the variation in the optimal length vs. variation in the mirror RoC is roughly linear within the ±1% RoC tolerance. So, we can choose two baseline mode definitions (one for each mirror topology) and then adjust as necessary following our RoC measurements.
For R = 2.5 m, the optimal length (see previous post) is LRT = 1.150 m, and the variation in this is dLRT/dR ~ +0.44 m/m.
Here is an illustration of the geometry:
The input q parameters, defined at the point over the edge of the OMC slab where the beam first crosses---(40mm, 150mm) on the OptoCad drawing---are, in meters:
For R = 2.5 m, the optimal length is LRT = 1.246 m, and the variation in this is also dLRT/dR ~ +0.44 m/m.
q parameters, defined as above:
Here is the proposed RoC measurement setup. Koji tells me that this is referred to as "Anderson's method".
We would like to use a linear cavity to measure the RoC of the curved mirrors independently (before forming the ring cavity), since the degeneracy of HOMs will make the fitting easier.
If we decided that the symmetric sidebands are too unwieldy, or that we have issues from sidebands on sidebands, we can accomplish the same style measurement using an AOM-shifted pickoff of the pre-PDH EOM beam. The advantage of the former method is that we don't have to use any polarization tricks.
Here is a more detailed version of the setup, so that we can gather the parts we will need.
EDIT (ZK): Koji points out that (1 - Ti) should really be the non-resonant reflectivity of the aligned cavity, which is much closer to 1. However, it should *actually* be the non-resonant reflectivity of the entire OMC assembly, including the steering mirror (see bottom of post). The steering mirror has T ~ 0.3%, so the true results are somewhere between my numbers and those with (1 - Ti) -> 1. In practice, though, these effects are swamped by the other errors.
More information about the power-dependent visibility measurement:
As a blanket statement, this measurement was done by exact analogy to those made by Sam and Sheon during S6 (c.f. LHO iLog 11/7/2011 and technical note T1100562), since it was supposed to be a verification that this effect still remains. There are absolutely better ways to do (i.e., ways that should give lower measurement error), and these should be investigated for our characterization. Obviously, I volunteer.
All measurements were made by reading the output voltages produced by photodetectors at the REFL and TRANS ports. The REFL PD is a BBPD (DC output), and the TRANS is a PDA255. Both these PDs were calibrated using a Thorlabs power meter (Controller: PM100D; Head: S12XC series photodiode-based---not sure if X = 0,2... Si or Ge) at the lowest and highest power settings, and these results agreed to the few-percent level. This can be a major source of error.
The power was adjusted using the HWP/PBS combination towards the beginning of the experiment. For reference, an early layout of the test setup can be seen in LLO:5978 (though, as mentioned above, the REFL and TRANS PDs have been replaced since then---see LLO:5994). This may or may not be a "clean" way to change the power, but the analysis should take the effect of junk light into account.
Below is an explanation of the three traces in the plot. First:
Now, the traces
The error bars in the measurement were dominated, roughly equally, by 1) systematic error from calibration of the PDs with the power meter, and 2) error from noise in the REFL_L measurement (since the absolute AC noise level in TRANS and REFL_L is the same, and TRANS >> REFL_L, the SNR of the latter is worse).
(1) can be helped by making ALL measurements with a single device. I recommend using something precise and portable like the power meter to make measurements at all the necessary ports. For REFL_L/UL, we can place a beam splitter before the REFL PD, and---after calibrating for the T of this splitter very well using the same power meter---both states can be measured at this port.
(2) can probably be helped by taking longer averaging, though at some point we run into the stability of the power setting itself. Something like 30-60s should be enough to remove the effects of the REFL_L noise, which is concentrated in the few-Hz region in the LLO setup.
One more thing I forgot was the finite transmission of the steering mirror at the OMC input (the transmission of this mirror goes to the QPDs). This will add a fixed error of 0.3%, and I will take it into account in the future.
I found that, in fact, I had lowered the modulation depth since when I measured it to be 0.45 rads --> Psb = 0.1.
Here is the sweep measurement:
This is Psb = 0.06 --> gamma = 0.35 rads.
This changes the "raw transmission" and "coupling", but not the inferred visibility:
I also measured the cavity AMTF at three powers today: 0.5 mW, 10 mW, and 45 mW input.
They look about the same. If anything, the cavity pole seems slightly lower with the higher power, which is counterintuitive. The expected shift is very small (~10%), since the decay rate is still totally dominated by the mirror transmissions even for the supposed high-loss state (Sam and Sheon estimated the roundtrip loss at high power to be ~1400 ppm, while the combined coupling mirrors' T is 1.6%). I have not been able to fit the cavity poles consistently to within this kind of error.
For various reasons, I had to switch NPROs (from the LightWave 126 to the Innolight Prometheus).
I installed the laser, realigned the polarization and modulation optics, and then began launching the beam into the fiber, though I have not coupled any light yet.
A diagram is below. Since I do not yet have the AOM, I have shown that future path with a dotted line. Since we will not need to make AMTFs and have a subcarrier at the same time, I have chosen to overload the function of the PBS using the HWP after the AEOM. We will operate in one of two modes:
One thing that concerns me slightly: the Prometheus is a dual-output (1064nm/532nm) laser, with separate ports for each. I have blocked and locked out the green path physically, but there is some residual green light visible in the IR output. Since we are planning to do the OMC transmission testing with a Si-based Thorlabs power meter---which is more sensitive to green than IR---I am somewhat worried about the ensuing systematics. I *think* we can minimize the effect by detuning the doubling crystal temperature, but this remains to be verified.
EDIT (ZK): Valera says there should be a dichroic beam splitter in the lab that I can borrow. This should be enough to selectively suppress the green.
Last night (Tuesday), I finished setting up and aligning most of the input optics for the OMC characterization setup. See the diagram below, but the setup consists of:
Today, we placed some lenses into the setup, in two places:
We (Koji, Lisa, and myself) had significant trouble getting more than ~0.1% coupling through the fiber, and after a while we decided to go to the 40m to get the red-light fiber illuminator to help with the alignment.
Using the illuminator, we realigned the input to the coupler and eventually got much better---but still bad---coupling of ~1.2% (0.12 mW out / 10 mW in). Due to the multi-mode nature of the illuminator beam, the output cannot be used to judge the collimation of the IR beam; it can only be used to verify the alignment of the beam.
With 0.12 mW emerging from the other end of the fiber, we could see the output quite clearly on a card (see photo below). This can tell us about the required input mode. From the looks of it, our beam is actually focused too strongly. We should probably replace the 75mm lens again with a slightly longer one.
Lisa and I concurred that it felt like we had converged to the optimum alignment and polarization, which would mean that the lack of coupling is all from mode mismatch. Since the input mode is well collimated, it seems unlikely that we could be off enough to only get ~1% coupling. One possibility is that the collimator is not well attached to the fiber itself. Since the Rayleigh range within it is very small, any looseness here can be critical.
I think there are several people around here who have worked pretty extensively with fibers. So, I propose that we ask them to take a look at what we have done and see if we're doing something totally wrong. There is no reason to reinvent the wheel.
My hypothesis about the input-side collimator turned out to be correct.
I removed the fiber from the collimator and mount at the input side, and then injected the illuminator beam from this side. Since we already saw a nice (but dim) IR beam emerging from the output side the other night, it followed that that collimator was correctly attached. With the illuminator injected from the input side, I also saw a nice, collimated red beam emerging from the output. So, the input collimator was not properly attached during our previous attempts, leading to the abysmal coupling.
The problem is that the mount does not allow you to remove and reattach the fiber while the collimator is already attached, and the dimensions make it hard to fit your fingers in to tighten the fiber to the collimator once the collimator is in the mount. I disassembled the mount and found a way to attach/reattach the fiber that preserves the tight collimator contact. I will upload a how-to shortly.
With this fix, I was able to align the input beam and get decent coupling:
EOM path: ~70%
AOM path: ~50%
We installed the MMT that matches the fiber output to the OMC on a 6"x12" breadboard. We did this so that we can switch from the "fauxMC" (OMC mirrors arranged with standard mounts for practice locking) to the real OMC without having to rebuild the MMT.
The solution that Koji found was:
z = 0: front face of the fiber output coupler mount
z = 4.8 cm: f = 35mm lens
z = 21.6 cm: f = 125mm lens
This should place the waist at z ~ 0.8 m. Koji has the exact solution, so I will let him post that.
The lenses are on ±0.5" single-axis OptoSigma stages borrowed from the TCS lab. Unfortunately, the spacing between the two lenses is very close to a half-integer number of inches, so I had to fix one of them using dog clamps instead of the screw holes to preserve the full range.
Koji also built the periscope (which raises the beam height by +1.5") using a vertical breadboard and some secret Japanese mounts. Part of it can be seen in the upper left corner of the photo below---sorry for not getting a shot of it by itself.
Then, I started to check the AOM path. I noticed that the 1st (or -1st) order beam is very weak.
The deflection efficiency is ~0.1%. Something is wrong.
I checked the driver. The driver's coupler output (1:10) show the amplitude ~1V. (good)
I check the main output by reducing the offset. When the coupler output is 100mV, the main output was 1V. (good)
So is the AOM itself broken???
As Koji noticed that the AOM efficiency was very low, I figured I would try looking at it with a fresh set of eyes. The end result is that I have to agree that the AOM appears to be broken.
First, I measured the input impedance of the AOM using the AG4395A with the impedance test kit (after calibrating). The plot is below. The spec sheet says the center frequency is 200 MHz, at which Zin should be ~50 ohms. It crosses 50 ohms somewhere near 235 MHz, which may be reasonable given that the LC circuit can be tuned by hand. However, it does surprise me that the impedance varies so much over the specified RF range of ±50 MHz. Maybe this is an indication that something is bad.
I removed the cover of the modulator (which I think Koji did, as well) and all the connections looked as I imagine they should---i.e., there was nothing obviously broken, physically.
I then tried my hand at realigning the AOM from scratch by removing and replacing it. I was not able to get better than 0.15%, which is roughly what Koji got.
So, perhaps our best course of action is to decide what we expect the Zin spectrum to look like, and whether that agrees with the above measurement.
Tonight, we locked the "fauxMC". We obtained a visibility of >99%.
Koji had aligned it roughly last night, but we wanted to have a couple steering mirrors in the path for this practice cavity (the periscope mirrors will serve this function in the real setup), so we marked the alignment with irises and installed two extra mirrors.
After obtaining flashes with the WinCam placed at the output coupler, we removed the WinCam and put a CCD camera at one of the curved mirror transmissions and used this to get a strong TEM00 flash. Then, we installed the REFL PD/CCD, swept the laser PZT and optimized the alignment by minimizing the REFL dips. Finally, we connected the RF electronics and locked the cavity with the LB box. We used whatever cables we had around to trim the RF phase, and then Koji made some nice SMA cables at the 40m.
One thing we noticed was that we don't have enough actuation range to keep the cavity locked for very long---even with the HV amp (100V). We are going to offload to the NPRO temperature using an SR560 or pomona box circuit. We may also make an enclosure for the cavity to protect it from the HEPA blasting.
Tomorrow, after we do the above things, we will practice measuring the transmission, length (FSR) and mode spectrum of the cavity before moving on to the real McCoy.
I’ve borrowed the black and decker toaster oven to dry some sonicated parts. It is temporarly located in the QIL lab.
I see that these measurements are done out to 100 kHz - I guess there is no reason to suspect anything at 55 MHz which is where this QPD will be reading out photocurrent given the low frequency behavior looks fine? The broad feature at ~80 kHz is the usual SR785 feature I guess, IIRC it's got to do with the display scanning rate.
The measured floor level of the dark current was below the shot noise level for the DC current of 0.1mA (i.e. 6pA/rtHz).
We have measured the dimensions and mass of the OMC glass plates/breadboards:
Note by Koji:
attachment 6: DCPD preamp looks like the opamp is wired for positive feedback?
are there any measurements of the BRDF of these things? I'm curious how much light is backscattered into the incoming beam and how much goes out into the world.
Maybe we can take some camera images of the cleaned ones or send 1-2 samples to Josh. No urgency, just curiosity.
I saw that ANU and also some labs in India use this kind of blue/green glass for beam dumps. I don't know much about it, but I am curious about its micro-roughness and how it compares to our usual black glass. For the BRDF, I think the roughnesss matters more for the blackness than the absorption.
I attempted to fit the data taken by Koji of the beam spot precession at the CCD in order to find the location of the curvature bottom in terms of its distance (d) and angle () from the centre of the mirror. This was done using the method described in a previous similar measurement and Section 2.1.3 of T1500060.
Initially, I attempted doing a circle_fit on python as seen in Attachment 1, and even though more points seem to coincide with the circle, Koji pointed out that the more appropriate way of doing it would be to fit the following function:
since that would allow us to measure the angle more accurately; is the anti-clockwise measured angle that the curvature bottom makes with the positive x direction.
As seen on the face of the CCD, x is positive up and y is positive right, thus, plotting it as the reflection (ref. Attachment 2) would make sure that is measured anti-clockwise from the positive x direction.
The distance from the curvature bottom is calculated as
r: radius of precession on CCD screen (value obtained from fit parameters, uncertainty in this taken from the std dev provided by fit function)
R: radius of curvature of the mirror
L: Distance between mirror and CCD
R = 2.575 0.005 m (taken from testing procedure doc referenced earlier) and L = 0.644 0.005 m (value taken from testing doc, uncertainty from Koji)
Using the measurements of PZTs 12,13 taken by Stephen, I estimated the wedging angle and orientation following Section 2.3.1 of T1500060. The results can be found in Attachment1 and is summarised as follows.
For PZT 12, PZT 13 respectively:
Avg. height = 2.0063 mm, 2.0035 mm
Wedge direction (from the same direction as in the doc: positive right) = 120 deg, 120 deg
Wedge angles = 45.8 arcsec, 30.6 arcsec
This was done assuming that the measurements were taken uniformly at intervals of 60deg along the inner rim of the PZT. The diameter (2r) of the inner rim, according to T1500060, is 9mm. The measured heights were fitted with the function
as depicted in Attachment2 to find wedging angle and orientation .
Wedge and thickness measurements of PZTs 12 and 13 took place after debonding and cleaning - results are shown in the first image (handwritten post-it format).
These thickness measurements seem to have come back thinner than previous measurements. It is possible that I have removed some PZT material while mechanically removing glue. It is also possible that there is systematic error between the two sets of measurements. I did not run any calculations of wedge ange or orientation on these data.
Note that cleaning of debonded PZTs involved mechanically separating glue from the planar faces of PZTs. The second image shows the razer blade used to scrape the glue away.
There were thick rings of glue where there had been excess squeezed out of the bond region, and there was also a difficult-to-remove bond layer that was thinner. I observed the presence of the thin layer by its reflectivity. The thick glue came off in patches, while the thin glue came off with a bit of a powdery appearance. It was hard to be certain that all of the thin bond layer came off, but I made many passes on each of the faces of the 2 PZTs that had been in the bonded CM assemblies. I found it was easiest to remove the glue in the bonded
I was anticipating that the expected 75-90 micron bond layer would affect the micrometer thickness measurements if it was still present, but I did not notice any irregularities (and certainly not at the 10 micron level), indicating that the glue was removed successfully (at least to the ~1 micron level).
Yesterday I measured the thickness of the PZTs in order to get an idea how much the PZTs are wedged.
For each PZT, the thickness at six points along the ring was measured with a micrometer gauge.
The orientation of the PZT was recognized by the wire direction and a black marking to indicate the polarity.
A least square fitting of these six points determines the most likely PZT plane.
Note that the measured numbers are assumed to be the thickness at the inner rim of the ring
as the micrometer can only measure the maximum thickness of a region and the inner rim has the largest effect on the wedge angle.
The inner diameter of the ring is 9mm.
The measurements show all PZTs have thickness variation of 3um maximum.
The estimated wedge angles are distributed from 8 to 26 arcsec. The directions of the wedges seem to be random
(i.e. not associated with the wires)
As wedging of 30 arcsec causes at most ~0.3mm spot shift of the cavity (easy to remember),
the wedging of the PZTs is not critical by itself. Also, this number can be reduced by choosing the PZT orientations
based on the estimated wedge directions --- as long as we can believe the measurements.
Next step is to locate the minima of each curved mirror. Do you have any idea how to measure them?
(In short, attachment 1 shows the two chosen sets of components and the configuration according which they must be bonded to minimize the total vertical angular deviation.)
The specfic components and configuration were chosen as follows, closely following Section 2.3.3 of T1500060:
Mounting prisms: 1,2,12,14,15 (Even though there is mention of M17 in the attachments, it can not be used because it was chipped earlier.)
Curved mirrors: 10,13
For a given choice of prism, PZT and mirror, the PZT can be placed either at 0deg or 180deg, and the mirror can rotated. This allows us to choose an optimal mirror rotation and PZT orientation which minimises the vertical deviation.
Total vertical angle
was measured by Koji as described in elog 369.
, are the wedge angle and orientation respectively and were measured earlier and shown in elog 373 .
, The measurement of the location of the curvature bottom (d, ) of the mirrors is shown in elog 372 . The optimal is to be found.
These steps were followed:
These are the ones that were chosen:
The method of attaching them is depicted in Attachment 1.